In this chapter we gathered the three main problems one faces in metal-forming simulation once the element and the constitutive formulation are chosen:

1. The contact problem is still among the most challenging ones, as it introduces a very stiff constraint, which results in a discontinuity of the velocity field at the onset of the contact.

2. The appropriate introduction of the friction law is also crucial in forming processes, as the material flow can be more sensitive to a small variation of the friction parameters than to a change of rheology.

3. The introduction in the finite-element formulation of the incompressibility requirement for dense materials, when elasticity is neglected. This topic is not straightforward from the engineering point of view, so it has motivated very theoretical works in the field of functional analysis; we shall not review these here as it is far beyond the scope of this book.

The Contact Problem

The contact problem with tools is probably the most challenging issue in modeling metal-forming processes. It is intrinsically a difficult mathematical problem, as it corresponds to a time discontinuity of the velocity when contact is established and, even in the stationary case, it involves unilateral conditions. Moreover if contact is not taken into account properly, or if the approximation is too crude, the final results of the computation are greatly affected and can be completely wrong. For simplicity we shall restrict ourselves here to rigid tools.